Theorem termco 49443

Description: The object of a terminal category. (Contributed by Zhi Wang, 17-Nov-2025.)

Hypotheses

Ref	Expression
termcbas.c	⊢ (𝜑 → 𝐶 ∈ TermCat)
termcbas.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
termco	⊢ (𝜑 → ∪ 𝐵 ∈ 𝐵)

Proof of Theorem termco

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		termcbas.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ TermCat)
2		termcbas.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3	1, 2	termcbas 49442	. 2 ⊢ (𝜑 → ∃𝑥 𝐵 = {𝑥})
4		unieq 4878	. . . . . 6 ⊢ (𝐵 = {𝑥} → ∪ 𝐵 = ∪ {𝑥})
5		unisnv 4887	. . . . . 6 ⊢ ∪ {𝑥} = 𝑥
6	4, 5	eqtrdi 2780	. . . . 5 ⊢ (𝐵 = {𝑥} → ∪ 𝐵 = 𝑥)
7		vsnid 4623	. . . . 5 ⊢ 𝑥 ∈ {𝑥}
8	6, 7	eqeltrdi 2836	. . . 4 ⊢ (𝐵 = {𝑥} → ∪ 𝐵 ∈ {𝑥})
9		id 22	. . . 4 ⊢ (𝐵 = {𝑥} → 𝐵 = {𝑥})
10	8, 9	eleqtrrd 2831	. . 3 ⊢ (𝐵 = {𝑥} → ∪ 𝐵 ∈ 𝐵)
11	10	exlimiv 1930	. 2 ⊢ (∃𝑥 𝐵 = {𝑥} → ∪ 𝐵 ∈ 𝐵)
12	3, 11	syl 17	1 ⊢ (𝜑 → ∪ 𝐵 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {csn 4585  ∪ cuni 4867  ‘cfv 6499  Basecbs 17155  TermCatctermc 49434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-iota 6452  df-fv 6507  df-termc 49435

This theorem is referenced by: (None)